Sunday, January 27, 2008

Let's pay a return visit to Brenna’s 1994 paper (Curve Fitting for Restoration of Accuracy for Overlapping Peaks …), which we first looked at in November.

Why are we doing this?

Well this paper formed the basis for a clear difference of opinion between Brenna and Meier-Augenstein during the hearing. All of Floyd’s IRMS F3 chromatograms showed a degree of overlap between the 5bP and 5aP peaks. If this resulted in an unknown error and that error biased the result in a more negative direction, it would/should have put those results in question (more negative equates to higher probability that testosterone was not generated by the athlete’s body).

Meier-Augenstein argued that the error was of unknown magnitude and in a more negative direction. Brenna argued that the error was very small and in a less negative error (i.e. the error actually made it look less probable that the athlete doped).

Meier-Augenstein’s opinion no doubt comes from his years of work and research in this discipline. We can trace Brenna’s opinion back to a paper he published which, amongst other things, detailed his observations when peaks of almost identical carbon composition overlap. His observations were that a systematic error occurs when peaks overlap and that error (not small, by the way) made the CIR for the substance being investigated look less negative than it really was.

What he didn’t investigate in that paper was why that happened. Understanding results is usually a prerequisite to applying them to different situations. Brenna chose not to take that precaution in his testimony.

The GC/C/IRMS process separates the C12 and C13 isotopes and produces two individual signals, the m44 and the m45 peaks (corresponding to the C12 and C13 isotopes, respectively). The relative size of these two signals is proportional to the carbon isotope ratio (CIR) of the substance under investigation. The CIR is actually calculated as the area of the m45 peak divided by the area of the m44 peak, using integration. In an ideal world, the m45 peak can be thought of a scaled down version of the m44 peak (100’s of times smaller). In practice, these two peaks do not align on the time axis perfectly, the error is very small (typically 150 milliseconds for a peak that may be approximately 35 seconds wide at the base).

Brenna mentions in his paper that his software corrects the misalignment of m44 and m45. He also states that he uses the “perpendicular drop” method to separate peaks (a vertical line positioned at the minima of the intersection of the two peaks and drawn down to the background level). Can our spreadsheet cope with these demands?. Darned right it can !

The goal of this first part is to recreate Brenna’s observations, so let’s dust off our spreadsheet and fire her up. We’re going to define two peaks, each with a d45 value of –27. We’re going to set the time difference between m45 and m44 to exactly zero and then make sure we’re measuring them accurately. Figure 1 shows the output. Two peaks, no overlap, both reporting a correct d45 of -27. [note: d45 defines the CIR of a substance relative to an international standard in parts per 1000, so –27 reads as a CIR which is –27 parts per thousand less than the standard].

Figure 1. Exact m45 and m44 alignment. No overlap between peaks.

It’s worth noting here that we only show the m44 trace, as appears to be common practice when presenting these results, but remember that there’s another trace (the m45) not shown.

Right, let’s introduce some overlap. Figure 2 shows the results we get. Note that we’re still measuring the true value. No error there.

OK, in our ideal world with two equal peaks, both having perfect peak responses and perfectly aligned m45 and m44 responses, it would appear that overlap has no effect on our results.

The question now, is what factors were at work to produce the results that Brenna observed?

Let’s go back to the drawing board. We know that m45 tends to lead m44 so we’ll scrap our assumption of perfect peak alignment and allow the m45 response to lead the m44 response by some nominally small value, say 100 ms.

Figure 4 shows our two peaks again. Same situation as Figure 2, but with the m45 response leading the m44 response. Now we’re seeing something change. The measured value is wrong and it is more negative (remember, the true value of our peaks are both -27)

Figure 4. m45 leads m44 by 100ms. Moderate overlap between peaks.

Figure 5 shows an increased overlap. Same situation as Figure 3, but with the m45 response leading the m44 response. The overlap has increased and so has our error. This is starting to resemble Brenna’s research, apart from one major difference – our error is going in the opposite direction to his observations.

Having the m45 response lead the m44 response gives the results that Meier-Augenstein predicted – an error which makes the results appear more negative. That was also his stated reason for the response he predicted. He must have observed systems where the m45 signal lead the m44 signal by some finite amount.

Figure 5. m45 leads m44 by 100ms. Significant overlap between peaks.

Right, let’s review what we’ve observed so far. In our ideal world, with no other sources of error other than the alignment of the m45 and m44 responses, perfect alignment of m45 and m44 results in no error when peaks of identical CIR overlap. We can infer that Brenna could not have had perfect alignment of the m45 and m44 signals. Having the m45 lead the m44 results in a systematic error, proportional to degree of overlap, which biases the result in a more negative direction.

Let’s see what happens if we have m45 response lagging the m44 response by some small amount, again 100ms. Figure 6 shows the same situation as Figure 4, but with the m45 response lagging the m44 response. Right, now we’re getting somewhere. An error which makes the result less negative. So far so good.

Figure 6. m45 lags m44 by 100ms. Moderate overlap between peaks.

Figure 7 shows an increased overlap. Same situation as Figure 5, but with the m45 response lagging the m44 response. The overlap has increased and so has our error. This looks just like the results Brenna observed. A systematic error which is proportional to degree of overlap and results in a less negative bias.

Figure 7. m45 lags m44 by 100ms. Significant overlap between peaks.

OK, so we can provide an explanation which satisfies both Brenna’s observations and Meier-Augenstein’s predictions. It’s all to do with the relative positioning of the m45 and m44 responses. Perfect alignment and there’s no error. A small misalignment (in our case, approximately 0.3% of the peak width) and a systematic error is observed proportional to degree of overlap and biased in either the more negative or less negative direction (depending on whether m45 leads or lags m44).

This is probably a good point to establish some ground rules. Perfect alignment is unachievable. This is not an analogue system with continuous signals. It is a digital system which samples the m44 and m45 responses at some fixed frequency. Regardless of what algorithm is used by the software to try and align the two signals there will always be a finite error in that process. There are many other factors which make perfect alignment of m45 and m44 an impossibility in practice, e.g. slightly different peak widths, slightly different peak shapes, etc. All these factors conspire to make the job of the software a demanding one.

The next part will demonstrate why we observe these results and what we can infer from the LNDD results.

8
comments:

It is probably a lot of work, but it sure would be clearer if you added a visual representation of the m45 peak, so people (especially people new to this) can clearly see that it is the slightly misaligned m44 and m45 peaks that make the CIR more or less negative.

And isn't the misalignment always in one direction or the other (on the x-axis)? Doesn't the m45 always precede the m444?

Let's use your Figure 4. You're looking at two peaks, and each peak combines the m44 and m45 response. Why would you do your analysis this way, by combining the two responses into a single peak? As I understand the IRMS machine, the IRMS has a separate collector for the m44 and m45 response. Isn't it possible to separately graph the m44 and m45 response?

Part 2 will answer your questions, but I'll provide a brief answer now so that you can dive straight into it when it arrives.

Mike, part 2 shows the m45 and m44 peaks. Part 1 just showed the m44 because that was the format of Brenna's paper (and the LNDD results). The natural misalignment is always in one direction. It's what happens after that, when the software attempts to compensate for that difference that matters. That's what we investigat in Part 2.

Larry, we havn't combined the responses of both m45 and m44, we've only shown the m44 response in Part 1. We were just trying to establish the cause of the error. Part 2 will show the mechanism and how m44 and m45 react to give the final result.

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TBV is personally biased towards Floyd. I think it'll be a better world if he proves his innocence, and some inquisitors meet their own just ends. Interspersed between daily link roundups are pieces of commentary slanted towards understanding what will prove innocence in the discipline proceeding, and what will rehabilitate his reputation in the public eye. Make of them what you will. Agreement with me is not required, though I am right.